A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is a fundamental concept in mathematics, particularly in linear algebra, and has numerous applications in various fields such as physics, engineering, computer science, and economics.

Key points about matrices:

  1. Dimensions: A matrix is defined by its number of rows (m) and columns (n), denoted as an m × n matrix.

  2. Elements: Each entry in a matrix is called an element or component, typically denoted as a_ij, where i represents the row and j represents the column.

  3. Square matrix: A matrix with an equal number of rows and columns (m = n) is called a square matrix.

  4. Matrix operations: a. Addition: Two matrices of the same dimensions can be added element-wise. b. Subtraction: Similar to addition, subtraction is performed element-wise. c. Multiplication: Matrix multiplication is a more complex operation, where the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix has the number of rows of the first matrix and the number of columns of the second matrix.

  5. Identity matrix: A square matrix with 1s on the main diagonal and 0s elsewhere is called an identity matrix, denoted as I_n for an n × n matrix.

  6. Transpose: The transpose of a matrix A, denoted as A^T, is obtained by interchanging the rows and columns of the original matrix.

  7. Inverse: A square matrix A has an inverse, denoted as A^-1, if the product of A and A^-1 equals the identity matrix. Not all matrices have an inverse.

  8. Determinant: The determinant is a scalar value associated with a square matrix, often used to determine if a matrix is invertible (non-zero determinant) and to calculate the inverse.

Matrices have numerous applications, such as solving systems of linear equations, transforming coordinates in computer graphics, representing graphs and networks, and analyzing data in machine learning and statistics.